Photographs of the freed families released by the aid group Norwegian Refugee Council (NRC), which runs a camp for internally displaced people 20 miles outside Fallujah, showed exhausted and malnourished-looking children in dirty clothes.

One family escaped in the middle of the night. “They took off their slippers to make less noise. They hid in big drainage pipes, before running to the border raising white flags made of cloth,” NRC said.

A mother said there was little left to eat in the besieged city: “We have survived for months on just dates and by drinking water from the river.”

Aid agencies told the Telegraph they cannot get access to the city, which no longer has electricity or clean water supplies.

Nasr Muflahi, NRC country director in Iraq, said: “The stories coming out of Fallujah are horrifying. People who managed to flee speak of extreme hunger and starvation.

“We cannot verify this as the centre is inaccessible; we haven’t been able to assist people inside Fallujah for months and we don’t know the full extent of the catastrophe unfolding there. Now they are caught in the crossfire with no safe way out,” he said.